Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices

The photonic bands of two-dimensional triangular and square lattices composed of circular rods were classified by means of the group theory based on the symmetry of the lattice structure. According to this classification, it was shown that the uncoupled mode, or the mode that cannot be excited by an external plane wave, which we previously found for the triangular lattice by the numerical calculation of the transmittance, is an antisymmetric mode under the relevant mirror reflection, and this fact is consistent with the observation by Robertson et al. It was also shown that triangular and square lattices with ${\mathit{C}}_{6\mathit{v}}$ or ${\mathit{C}}_{4\mathit{v}}$ symmetry have many other uncoupled modes with relatively low eigenfrequencies and some of them can be easily identified as the spectral ranges of total reflection in spite of their nonzero state density.